Question: Solve for $x$ : $ 5|x + 9| - 5 = 3|x + 9| + 6 $
Explanation: Subtract $ {3|x + 9|} $ from both sides: $ \begin{eqnarray} 5|x + 9| - 5 &=& 3|x + 9| + 6 \\ \\ { - 3|x + 9|} && { - 3|x + 9|} \\ \\ 2|x + 9| - 5 &=& 6 \end{eqnarray} $ Add ${5}$ to both sides: $ \begin{eqnarray} 2|x + 9| - 5 &=& 6 \\ \\ { + 5} &=& { + 5} \\ \\ 2|x + 9| &=& 11 \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{2|x + 9|} {{2}} = \dfrac{11} {{2}} $ Simplify: $ |x + 9| = \dfrac{11}{2}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 9 = -\dfrac{11}{2} $ or $ x + 9 = \dfrac{11}{2} $ Solve for the solution where $x + 9$ is negative: $ x + 9 = -\dfrac{11}{2} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& -\dfrac{11}{2} \\ \\ {- 9} && {- 9} \\ \\ x &=& -\dfrac{11}{2} - 9 \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $2$ $ x = - \dfrac{11}{2} {- \dfrac{18}{2}} $ $ x = -\dfrac{29}{2} $ Then calculate the solution where $x + 9$ is positive: $ x + 9 = \dfrac{11}{2} $ Subtract ${9}$ from both sides: $ \begin{eqnarray} x + 9 &=& \dfrac{11}{2} \\ \\ {- 9} && {- 9} \\ \\ x &=& \dfrac{11}{2} - 9 \end{eqnarray} $ Change the ${ - 9}$ to an equivalent fraction with a denominator of $2$ $ x = \dfrac{11}{2} {- \dfrac{18}{2}} $ $ x = -\dfrac{7}{2} $ Thus, the correct answer is $x = -\dfrac{29}{2} $ or $x = -\dfrac{7}{2} $.